There are several prior art techniques for measuring the distribution of the particle size in a sample by using light scattering. Generally, to measure the sizes of individual particles, for example, in a flowing stream of a liquid or gas, the particle-containing sample is illuminated by a constant light source and the intensity of light scattered by the particle is detected. A particle scatters the light by an amount directly related to the particle size; in general, bigger particles scatter more light than smaller particles. The relation between the amount of scattering and particle size may be determined either from theoretical calculations or through calibration process. With knowledge of this relation, for a single particle at a time, the detected scattered light intensity provides a direct measure of the particle size. The distribution of particle sizes in a sample may be determined by individually passing each particle in the sample, or a suitable portion of the sample, through the scattered light detection apparatus and tabulating the sizes of the various particles. In practice, this method is generally restricted to particles greater than 0.5 microns. Moreover, this method is relatively slow since particles must be detected individually. This technique is referred to in the prior art as optical particle counting (OPC).
A second prior art technique of particle sizing by light scattering is referred to as static or "classical" light scattering (CLS). This method is based upon illumination of a sample containing the particles-to-be-sized followed by the measurement of the intensity of scattered light at several predetermined angles. Because of intra-particle destructive interference, the intensity of light scattered from a particle depends on both the size and composition of the particle and the angle at which the measurement is made. This method of particle sizing based on the angular dependence of the scattered intensity can be used to measure the size distribution of a group of particles, as opposed to the first method noted above which is restricted to individual particles.
To implement the CLS measurement method, a sample of particles dispersed in a fluid is illuminated along an input axis, and the intensity of scattered light is measured at several predetermined angles. The scattered light intensity at each angle may be measured simultaneously with a multitude of detectors or consecutively, by moving a single detector around the sample to permit measurement of the intensity at each desired angle.
For large particles, for example, having diameters greater than 1 micron, the scattered light flux is concentrated in the forward direction relative to the input axis. Instruments for sizing large particles are referred to as laser diffraction devices. For sizing of smaller particles, for example, having diameters as low as 0.2 microns, the scattered light flux has significant magnitude both at lower and higher scattering angles relative to the input axis. The angular intensity measurements used on smaller particles are termed total integrated, or average, intensity measurements and may be displayed in a form known as Zimm plots.
A third prior art technique for particle sizing by light scattering is dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS) or quasi-elastic light scattering (QELS). See B. E. Dahneke, "Measurement of Suspended Particles by Quasi-Elastic Light Scattering," John Wiley & Sons, Inc., New York, 1983. This technique is based on measuring the time-fluctuations of the intensity of light scattered from an illuminated sample containing a group of particles which are diffusing through a fluid, that is, randomly moving due to collisions with solvent molecules and other particles. For example, the particles may be macromolecules dissolved in a liquid, where the macromolecules may be ionized by the loss of a small number of charged atoms.
In accordance with the DLS technique, scattered light intensity is measured as a function of time at a selected angle with respect to an illumination input axis. The light intensity detected at any instant at the detector is dependent on the interference between the light scattered from each illuminated particle in the scattering volume. As the particles randomly diffuse through the solution, the interference of the light scattered from them changes and the intensity at the detector therefore fluctuates. Since smaller particles diffuse faster, the fluctuations resulting from the motion of relatively small particles vary faster than those resulting from the motion of larger particles. Thus, by measuring the time variation of the scattered light fluctuations at the detector, information representative of the distribution of particle sizes is available. More particularly, the autocorrelation function of the measured intensity is related to the distribution of particle sizes in the fluid. Conventional DLS instruments such as the Model N4 photon correlation spectrometer manufactured by Coulter Electronics, Inc., Hialeah, Fla., provide autocorrelation signals for the detected intensity suitable for measuring distributions including particle sizes as low as 0.003 microns. Accordingly, such devices have a size measuring range extending considerably below the above-noted individual particle and CLS methods.
Particle sizing measurements by the known DLS techniques are generally made in the following manner. The particles-to-be-sized are suspended or dissolved in a fluid, forming a sample. The sample is illuminated by a laser beam directed along an input axis. Although a laser is generally used to generate the beam, a non-coherent light source may alternatively be used.
The light scattered from the particles in the sample is detected by a photodetector, such as a photomultiplier, which is positioned at a predetermined angle. The particular angle may be selected by the operator, but usually only one angle is measured at a time. The photodetector produces a signal which varies with time as the scattered light intensity incident on the photodetector varies. This time-varying signal is applied to an autocorrelator analyzer, to compute the autocorrelation function of the photodetector signal. Typically, the autocorrelator computes the value of the autocorrelation function of the detected scattered light at as many as one hundred discrete time points. This autocorrelation function contains the information about the fluctuations in the detected scattered light, from which information about the distribution of particle sizes in the sample can be extracted. Thus, the autocorrelation function (acf) is the raw data of a DLS measurement. While most conventional DLS measurements are performed using this autocorrelation step, it is known that the acf of the intensity signal corresponds to the Fourier Transform of the power spectrum of that signal. Accordingly, a spectrum analyzer may be used in place of the autocorrelator to generate a power spectrum signal including the same information representative of the particle size distribution as is resident in the autocorrelation function. The frequency domain information resident in the power spectrum signal can be used to determine the particle size distribution.
In the prior art, there are several techniques for extracting the particle size distribution from the acf. For use with these techniques, the relationship between the acf and the size distribution can be expressed as: EQU g(t)=K(x(r)) (1)
where g(t) is the acf (or a function closely related to the acf), x(r) is the sought distribution of particle sizes (x is a function of r, the particle radius), and K is a function (or operator, linear or non-linear) which relates particle size to the acf. Thus, given the exact form of K, the autocorrelation function resulting from any distribution, x(r), of particle sizes would be known.
Since the acf, g(t), is what is actually measured in practice, the above relation must be inverted to yield the particle size distribution: EQU x(r)=K.sup.-1 (g(t)) (2)
Accordingly, for the measured acf for a sample of particles, the size distribution for those particles can be extracted by applying the operator K.sup.-1 to the measured acf, g(t). The operator K.sup.-1 is the generalized inverse of the operator K. In practice however, the acf is "ill-conditioned" so that the inversion process is generally difficult and complex, although there are a number of known techniques for performing the inversion.
An example of the form K.sup.-1 for one particular commonly used prior art extraction technique is: EQU x=(K.sup.t K+.alpha.H).sup.-1 K.sup.t q (3)
In this example, x is a vector whose components are the proportions of the particles of each size, g is a vector whose components are the values of the acf at different points in time, as computed by the autocorrelator, K is a matrix relating x to g, and H is a matrix which increases the conditioning of the inversion. K.sup.t is the transpose of the matrix K. Alpha (.alpha.) controls the amount of conditioning imposed on the solution. The inverse operator K.sup.-1 in this case can be written K.sup.- =(K.sup.t K+.alpha.H).sup.-1 K.sup.t, where alpha (.alpha.) is a smoothing parameter determined conventionally. The inversion is usually performed along with some non-negativity constraints imposed on the solution; these constraints are formally part of the inverse operator K.sup.-1. Other known methods for inversion are the histogram method, the singular value decomposition method, the delta function method, and the cubic spline method.
The size distribution, x(r), obtained from this extraction or "inversion" process can be expressed either as a continuous distribution as implied by (r), where the distribution is defined for particles of any size, or as a discrete size histogram expressed by the vector x where the distribution of particle sizes is defined at only a set number of particle sizes. The vector x is representative of a group of numbers (x(r.sub.1), x(r.sub.2), . . . , x(r.sub.n)) giving the relative proportion of scattered light intensity from particles of size r.sub.1, r.sub.2,. . .,r.sub.n, respectively. The size distribution (r) is referred to as a size histogram x herein below.
The size distribution x(r) and size histogram, x, are "intensity weighted" functions since these are representative of the relative proportion of particles as characterized by the relative amount of scattering intensity from particles of each size. However, these intensity weighted functions are dependent on the angle at which the measurement of scattered light was made. That is, the apparent proportion of particles of each size, as evidenced by the scattered light intensity contribution of particles of different sizes, depends on the angle at which the measurement is made. Thus, size distributions made at different angles cannot be directly compared using the intensity weighted distribution x(r) or histogram x.
Accordingly, if the amount of light scattered per particle, as a function of the scattering angle, is known, either through theoretical calculations or by an empirical method, the intensity weighted size distributions x(r) and histogram x at each angle can be directly compared by first converting those functions to corresponding mass, volume, or number weighted size distributions. For example, the intensity weighted histogram x may be converted in accordance with: EQU V=Cx
In this expression, v is the mass, volume or number weighted size histogram and C is the conversion matrix between the intensity weighted histogram, x, and the mass, volume or number weighted histogram, v. Similarly, the size distribution x(r) may be converted into a corresponding mass, volume or number distribution function v(r). Since all of these converted histograms and distribution functions provide the desired angle-independent information about the size distribution particles, they are referred to generally below as v and v(r), respectively.
A volume weighted histogram and distribution function provide a measure of the proportion of the total volume of particles in a sample as a function of particle size. For example, 50% of the volume of a sample of particles might come from particles of size 0.1 micron and the remaining 50% from particles of size 0.3 microns. Similarly, the mass weighted histograms and distributions provide a measure of the mass of particles in a sample as a function of size and the number weighted histograms and distribution of the numbers of particles in a sample as a function of size. For particles of the same density, the mass and volume weighted histograms and functions are the same. Vblume, mass and number weighted size histograms and distributions are generally more useful than the corresponding intensity weighted size histograms or distributions since the former relate to quantities which can be directly measured by other means.
All of the prior art light scattering measurement techniques are characterized by low resolution and poor reproducibility, the principal drawbacks of such methods. With respect to DLS sizing measurements, efforts have been made to try to increase the resolution. The general methods used to increase resolution either attempt to improve the signal-to-noise ratio of the measurement by collecting intensity data over a long period or over a large number of short periods and then averaging the results, or by using intensity data collected at several angles.
With the latter method, the data collected at different angles are substantially independent, and therefore data collected at one angle complement those collected at other angles. For example, data collected at lower scattering angles are generally more sensitive to the presence of large particles in the sample while, conversely, data collected at large scattering angles are more sensitive to the presence of smaller particles. A sample containing both large and small particles can therefore be accurately sized by using the data from two or more angles, where relatively lower angle or angles provide information about the larger particles and relatively high angle or angles provide information about the smaller particles. In contrast, measurement at a single low angle would provide relatively little and possibly obscured information about the smaller particles and hence the sizing resolution would be poor.
The prior art method of using several angles to enhance the sizing resolution involves simply making measurements at two or more angles and averaging the volume weighted histograms resulting from the measurements made at the two or more angles. Symbolically, the process of combining information obtained at several scattering angles by averaging results can be expressed by: ##EQU1## where the subscripts 1,2, . . . m refer to measurements made at the scattering angles .theta..sub.1 through .theta..sub.m. The inclusion of .theta. as an argument of the operator K.sup.-1 indicates that the inversion process, that is, the operator K.sup.-1, depends on the scattering angles. Each of the m intensity weighted histograms, x.sub.1, . . . ,x.sub.m, may be converted to an angle-independent volume weighted histogram, v.sub.1, . . . , v.sub.m, and then the m volume weighted histograms may be averaged to produce the "enhanced" resolution result, v: EQU v=(1/m)[v.sub.1 +v.sub.2 + . . . +v.sub.m ]
However, this volume weighted distribution, v, is not necessarily the solution which is the best fit to all the data. The size resolution obtainable for a single measurement at a single angle is quite low and the presence of particles of some sizes may not be detected at some angles. Thus, even when the intensity histograms are converted to volume histograms, the histograms obtained at different angles may give very different and apparently contradictory information.
It is an object of the present invention to provide an improved apparatus and method for measuring the distribution of particle sizes dispersed in a fluid.
Another object is to provide a particle sized distribution measuring apparatus and method characterized by relatively high resolution.